From: Salvador Flores <localadm@localadm-desktop.(none)>
Date: Fri, 16 Oct 2015 19:02:26 +0000 (-0300)
Subject: Solve p-laplacian elliptic problems for large p
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=d58329080d6370eaed4401c0768f28430b7c8fbe;p=code-gallery.git

Solve p-laplacian elliptic problems for large p
---

diff --git a/Elastoplastic Torsion/CMakeLists.txt b/Elastoplastic Torsion/CMakeLists.txt
new file mode 100644
index 0000000..82fb113
--- /dev/null
+++ b/Elastoplastic Torsion/CMakeLists.txt	
@@ -0,0 +1,40 @@
+##
+#  CMake script for the Elastoplastic Torsion problem
+#   at the deal.ii code-gallery 
+##
+
+# Set the name of the project and target:
+SET(TARGET "ElastoplasticTorsion")
+
+# Declare all source files the target consists of. Here, this is only
+# the one step-X.cc file, but as you expand your project you may wish
+# to add other source files as well. If your project becomes much larger,
+# you may want to either replace the following statement by something like
+#    FILE(GLOB_RECURSE TARGET_SRC  "source/*.cc")
+#    FILE(GLOB_RECURSE TARGET_INC  "include/*.h")
+#    SET(TARGET_SRC ${TARGET_SRC}  ${TARGET_INC}) 
+# or switch altogether to the large project CMakeLists.txt file discussed
+# in the "CMake in user projects" page accessible from the "User info"
+# page of the documentation.
+SET(TARGET_SRC
+  ${TARGET}.cc
+  )
+
+# Usually, you will not need to modify anything beyond this point...
+
+CMAKE_MINIMUM_REQUIRED(VERSION 2.8.8)
+
+FIND_PACKAGE(deal.II 8.0 QUIET
+  HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR}
+  )
+IF(NOT ${deal.II_FOUND})
+  MESSAGE(FATAL_ERROR "\n"
+    "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n"
+    "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n"
+    "or set an environment variable \"DEAL_II_DIR\" that contains this path."
+    )
+ENDIF()
+
+DEAL_II_INITIALIZE_CACHED_VARIABLES()
+PROJECT(${TARGET})
+DEAL_II_INVOKE_AUTOPILOT()
diff --git a/Elastoplastic Torsion/ElastoplasticTorsion.cc b/Elastoplastic Torsion/ElastoplasticTorsion.cc
new file mode 100644
index 0000000..1cec33a
--- /dev/null
+++ b/Elastoplastic Torsion/ElastoplasticTorsion.cc	
@@ -0,0 +1,1343 @@
+/* ---------------------------------------------------------------------
+ *
+ * Copyright (C) 2010 - 2015 by the deal.II authors 
+ *  							and Salvador Flores.
+ *                      
+ *
+ * 
+ * This is free software; you can use it, redistribute
+ * it, and/or modify it under the terms of the GNU Lesser General
+ * Public License as published by the Free Software Foundation; either
+ * version 2.1 of the License, or (at your option) any later version.
+ * The full text of the license can be found in the file LICENSE at
+ * the top level of the deal.II distribution.
+ *
+ * ---------------------------------------------------------------------
+ *
+ * Author: Salvador Flores, 
+ *         Center for Mathematical Modelling,
+ *		   Universidad de Chile, 2015.
+ */
+ 
+ 
+ /* 
+  This piece of software solves the elliptic p-laplacian 
+  boundary-value problems:
+  
+    Min {∫ 1/2 W(|Du|²)+ 1/p |Du|^p -fu : u=g on ∂S }   (1)
+     u
+ 
+  for large values of p, which approximates (see Alvarez & Flores 2015)
+  
+    Min {∫ 1/2 W(|Du|²) -fu : |Du|<1 a.s. on S, u=g on ∂S }   
+     u
+
+  By default W(t)=t and S=unit disk.
+  
+  Large portions of this code are borrowed from the deal.ii tutorials 
+  
+		     step-15, step-29. 
+ 	
+  For further details see the technical report available at
+  the documentation andmof	 at http://www.dim.uchile.cl/~sflores.
+   
+ */
+
+// Include files 
+
+#include <deal.II/base/quadrature_lib.h>
+#include <deal.II/base/function.h>
+#include <deal.II/base/logstream.h>
+#include <deal.II/base/utilities.h>
+#include <deal.II/base/convergence_table.h>
+#include <deal.II/base/smartpointer.h>
+#include <deal.II/base/parameter_handler.h>
+#include <deal.II/base/timer.h>
+
+#include <deal.II/lac/vector.h>
+#include <deal.II/lac/full_matrix.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/compressed_sparsity_pattern.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/precondition.h>
+#include <deal.II/lac/constraint_matrix.h>
+
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/grid_generator.h>
+#include <deal.II/grid/tria_accessor.h>
+#include <deal.II/grid/tria_iterator.h>
+#include <deal.II/grid/tria_boundary_lib.h>
+#include <deal.II/grid/grid_refinement.h>
+#include <deal.II/grid/grid_in.h>
+
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_accessor.h>
+#include <deal.II/dofs/dof_tools.h>
+#include <deal.II/dofs/dof_renumbering.h>
+
+#include <deal.II/fe/fe_values.h>
+#include <deal.II/fe/fe_q.h>
+
+#include <deal.II/numerics/vector_tools.h>
+#include <deal.II/numerics/matrix_tools.h>
+#include <deal.II/numerics/data_out.h>
+#include <deal.II/numerics/error_estimator.h>
+#include <deal.II/numerics/solution_transfer.h>
+
+#include <typeinfo>
+#include <fstream>
+#include <iostream>
+
+#include <deal.II/numerics/solution_transfer.h>
+
+// Open a namespace for this program and import everything from the
+// dealii namespace into it.
+namespace nsp
+{
+  using namespace dealii;
+
+// ********************************************************//
+ class ParameterReader : public Subscriptor
+{
+public:
+ParameterReader(ParameterHandler &);
+void read_parameters(const std::string);
+private:
+void declare_parameters();
+ParameterHandler &prm;
+};
+// Constructor
+ParameterReader::ParameterReader(ParameterHandler &paramhandler):
+prm(paramhandler)
+{}
+
+ void ParameterReader::declare_parameters()
+{
+
+prm.enter_subsection ("Global Parameters");
+{
+	prm.declare_entry("p", "100",Patterns::Double(2.1),
+		"Penalization parameter");
+	prm.declare_entry("known_solution", "true",Patterns::Bool(),
+		"Whether the exact solution is known");
+}
+prm.leave_subsection ();
+
+prm.enter_subsection ("Mesh & Refinement Parameters");
+{
+	prm.declare_entry("Code for the domain", "0",Patterns::Integer(0,2),
+		"Number identifying the domain in which we solve the problem");
+	prm.declare_entry("No of initial refinements", "4",Patterns::Integer(0),
+		"Number of global mesh refinement steps applied to initial coarse grid");
+	prm.declare_entry("No of adaptive refinements", "8",Patterns::Integer(0),
+		"Number of global adaptive mesh refinements");
+	prm.declare_entry("top_fraction_of_cells", "0.25",Patterns::Double(0),
+		"refinement threshold");
+	prm.declare_entry("bottom_fraction_of_cells", "0.05",Patterns::Double(0),
+		"coarsening threshold");
+}
+prm.leave_subsection ();
+
+
+prm.enter_subsection ("Algorithm Parameters");
+{
+	prm.declare_entry("Descent_direction", "0",Patterns::Integer(0,1),
+		"0: Preconditioned descent, 1: Newton Method");
+	prm.declare_entry("init_p", "10",Patterns::Double(2),
+		"Initial p");
+	prm.declare_entry("delta_p", "50",Patterns::Double(0),
+		"increase of p");
+	prm.declare_entry("Max_CG_it", "1500",Patterns::Integer(1),
+		"Maximum Number of CG iterations");
+	prm.declare_entry("CG_tol", "1e-10",Patterns::Double(0),
+		"Tolerance for CG iterations");
+	prm.declare_entry("max_LS_it", "45",Patterns::Integer(1),
+		"Maximum Number of LS iterations");
+	prm.declare_entry("line_search_tolerence", "1e-6",Patterns::Double(0),
+		"line search tolerance constant (c1 in Nocedal-Wright)");
+	prm.declare_entry("init_step_length", "1e-2",Patterns::Double(0),
+		"initial step length in line-search");
+	prm.declare_entry("Max_inner", "800",Patterns::Integer(1),
+		"Maximum Number of inner iterations");
+	prm.declare_entry("eps", "1.0e-8",Patterns::Double(0),
+		"Threshold on norm of the derivative to declare optimality achieved");
+	prm.declare_entry("hi_eps", "1.0e-9",Patterns::Double(0),
+		"Threshold on norm of the derivative to declare optimality achieved in highly refined mesh");
+	prm.declare_entry("hi_th", "8",Patterns::Integer(0),
+		"Number of adaptive refinement before change convergence threshold");
+}
+prm.leave_subsection ();
+
+}
+void ParameterReader::read_parameters (const std::string parameter_file)
+{
+declare_parameters();
+prm.read_input (parameter_file);
+}
+
+// ******************************************************************************************//
+// The solution of the elastoplastic torsion problem on the unit disk with rhs=4.
+
+template <int dim>
+class Solution : public Function<dim>
+{
+public:
+Solution () : Function<dim>() {}
+virtual double value (const Point<dim> &pto, const unsigned int component = 0) const;
+virtual Tensor<1,dim> gradient (const Point<dim> &pto, const unsigned int component = 0) const;
+};
+
+template <int dim>
+double Solution<dim>::value (const Point<dim> &pto,const unsigned int) const
+{
+double r=sqrt(pto.square());
+	if (r<0.5)
+	  return -1.0*std::pow(r,2.0)+0.75;          
+	else
+          return 1.0-r;
+}
+
+
+
+template <int dim>
+Tensor<1,dim> Solution<dim>::gradient (const Point<dim> &pto,const unsigned int) const
+{
+double r=sqrt(pto.square());
+	if (r<0.5)
+          return -2.0*pto;
+	else
+	  return  -1.0*pto/r;
+}
+
+
+
+
+// ****************************************************************************************** //
+/*                 Compute the Lagrange multiplier (as a derived quantity)                   */
+
+
+template <int dim>
+class ComputeMultiplier : public DataPostprocessor<dim>
+{
+  private:
+   	double p;
+  public:
+    ComputeMultiplier (double pe);
+
+    virtual
+    void compute_derived_quantities_scalar (
+                const std::vector< double > &  ,
+		const std::vector< Tensor< 1, dim > > & ,
+		const std::vector< Tensor< 2, dim > > &  ,
+		const std::vector< Point< dim > > &  ,
+		const std::vector< Point< dim > > & ,
+		std::vector< Vector< double > > &
+    ) const;
+
+    virtual std::vector<std::string> get_names () const;
+
+    virtual
+    std::vector<DataComponentInterpretation::DataComponentInterpretation>
+    get_data_component_interpretation () const;
+    virtual UpdateFlags get_needed_update_flags () const;
+};
+
+
+  template <int dim>
+  ComputeMultiplier<dim>::ComputeMultiplier (double pe): p(pe)
+     {}
+
+
+template <int dim>
+void ComputeMultiplier<dim>::compute_derived_quantities_scalar( 
+        	const std::vector< double > &  	/*uh*/,
+		const std::vector< Tensor< 1, dim > > &  duh,
+		const std::vector< Tensor< 2, dim > > &  /*dduh*/,
+		const std::vector< Point< dim > > &  /*	normals*/,
+		const std::vector< Point< dim > > &  /*evaluation_points*/,
+		std::vector< Vector< double > > &  	computed_quantities ) 	const
+{
+    const unsigned int n_quadrature_points = duh.size();
+
+    for (unsigned int q=0; q<n_quadrature_points; ++q)
+      { long  double sqrGrad=duh[q]* duh[q]; //squared norm of the gradient
+	long double exponent=(p-2.0)/2*std::log(sqrGrad);
+          computed_quantities[q](0) = std::sqrt(sqrGrad); // norm of the gradient
+          computed_quantities[q](1)= std::exp(exponent); // multiplier      
+}
+}
+
+
+
+
+
+template <int dim>
+std::vector<std::string>
+ComputeMultiplier<dim>::get_names() const
+{
+ std::vector<std::string> solution_names;
+solution_names.push_back ("Gradient norm");
+solution_names.push_back ("Lagrange multiplier"); 
+  return solution_names;
+}
+
+
+template <int dim>
+UpdateFlags
+ComputeMultiplier<dim>::get_needed_update_flags () const
+{
+  return update_gradients;
+}
+
+
+
+  template <int dim>
+  std::vector<DataComponentInterpretation::DataComponentInterpretation>
+ComputeMultiplier<dim>:: get_data_component_interpretation () const
+  {
+    std::vector<DataComponentInterpretation::DataComponentInterpretation>
+    interpretation;
+    // norm of the gradient
+    interpretation.push_back (DataComponentInterpretation::component_is_scalar); 
+	// Lagrange multiplier
+    interpretation.push_back (DataComponentInterpretation::component_is_scalar);
+    return interpretation;
+}
+
+
+
+
+
+// *************************************************************************************** //
+  template <int dim>
+  class ElastoplasticTorsion
+  {
+  public:
+    ElastoplasticTorsion (ParameterHandler &);
+    ~ElastoplasticTorsion ();
+    void run ();
+
+  private:
+    void setup_system (const bool initial_step);
+    void assemble_system ();
+    bool solve (const int inner_it);
+    void init_mesh ();
+    void refine_mesh ();
+    void set_boundary_values ();
+    double phi (const double alpha) const;
+    bool checkWolfe(double & alpha, double & phi_alpha) const;
+    bool determine_step_length (const int inner_it);
+    void print_it_message (const int counter, bool ks);
+    void output_results (unsigned int refinement) const;
+    void format_convergence_tables();
+    void process_solution (const unsigned int cycle);
+    void process_multiplier (const unsigned int cycle,const int iter,double time);
+    double dual_error () const;
+    double dual_infty_error () const;
+    double W (double Du2) const;
+    double Wp (double Du2) const;
+    double G (double Du2) const;
+
+
+        
+	ParameterHandler &prm;
+    Triangulation<dim>   triangulation;
+    DoFHandler<dim>      dof_handler;
+    ConstraintMatrix     hanging_node_constraints;
+    SparsityPattern      sparsity_pattern;
+    SparseMatrix<double> system_matrix;
+    ConvergenceTable convergence_table;
+    ConvergenceTable dual_convergence_table;
+    Vector<double>       present_solution;
+    Vector<double>       newton_update;
+    Vector<double>       system_rhs;
+    Vector<double>       grad_norm;
+    Vector<double>	      lambda;
+
+
+    double step_length,phi_zero,phi_alpha,phip,phip_zero;
+    double old_step,old_phi_zero,old_phip;
+	double L2_error;
+   double H1_error;
+   double Linfty_error; 
+   double dual_L1_error; 	
+   double dual_L_infty_error;
+    FE_Q<dim> fe;
+	double p;  
+    double line_search_tolerence; // c_1 in Nocedal & Wright
+    unsigned int dir_id;
+	std::string elements;
+	std::string Method;
+
+};
+
+/*******************************************************************************************/
+//                              Boundary condition
+
+  template <int dim>
+  class BoundaryValues : public Function<dim>
+  {
+  public:
+    BoundaryValues () : Function<dim>() {}
+
+    virtual double value (const Point<dim>   &p,
+                          const unsigned int  component = 0) const;
+  };
+
+
+  template <int dim>
+  double BoundaryValues<dim>::value (const Point<dim> &pto,
+                                     const unsigned int /*component*/) const
+  {	// could be anything else (theory works provided |Dg|_infty < 1/2)
+	return 0.0;	
+	
+	/* A challenging BC leading to overdetermined problems
+	   it is regulated by the parameter 0<eta<1.
+	   eta closer to 1 leads to more difficult problems.	
+
+	double pii=numbers::PI;     
+	double theta=std::atan2(p[1],p[0])+pii;
+	double eta=0.9;
+
+	    if (theta <= 0.5)
+			return eta*(theta*theta);
+		else if ((theta >0.5) & (theta<= pii-0.5))
+			return eta*(theta-0.25);
+		else if ((theta>pii-0.5)&(theta<= pii+0.5))
+			return eta*(pii-0.75-(theta-(pii-0.5))*(theta-(pii+0.5)));
+		else if ((theta>pii+0.5)&(theta<= 2*pii-0.5))
+			return eta*((2*pii-theta)-0.25);
+		else
+			return eta*((theta-2*pii)*(theta-2*pii) );*/
+  }
+
+
+
+/******************************************************************************/
+//                       Right-Hand Side
+template <int dim>
+class RightHandSide : public Function<dim>
+{
+	public:
+		RightHandSide () : Function<dim>() {}
+		virtual double value (const Point<dim> &p, 
+		const unsigned int component = 0) const;
+};
+
+template <int dim>
+double RightHandSide<dim>::value (const Point<dim> &p,
+const unsigned int /*component*/) const
+{ // set to constant = 4, for which explicit solution to compare exists
+  // could be anything
+	double return_value = 4.0;
+	return return_value;
+}
+
+
+
+/*******************************************************************/
+// The ElastoplasticTorsion class implementation
+
+ // Constructor of the class 
+  template <int dim>
+ElastoplasticTorsion<dim>::ElastoplasticTorsion (ParameterHandler &param):
+	prm(param),
+    dof_handler (triangulation),
+    L2_error(1.0),
+    H1_error(1.0),
+    Linfty_error(1.0),
+    dual_L1_error(1.0), 	
+    dual_L_infty_error(1.0),
+	fe(2)
+  {
+    prm.enter_subsection ("Global Parameters");
+	p=prm.get_double("p");
+    prm.leave_subsection ();
+    prm.enter_subsection ("Algorithm Parameters");
+	line_search_tolerence=prm.get_double("line_search_tolerence");
+	dir_id=prm.get_integer("Descent_direction");
+    prm.leave_subsection ();
+   	if (fe.degree==1)
+		 elements="P1";
+	else elements="P2";
+			
+	if (dir_id==0)
+	  Method="Precond";
+	else 
+	  Method="Newton";
+}
+
+
+
+template <int dim>
+ElastoplasticTorsion<dim>::~ElastoplasticTorsion ()
+  {
+    dof_handler.clear ();
+  }
+
+/*****************************************************************************************/
+//  print iteration message
+
+template <int dim>
+void ElastoplasticTorsion<dim>::print_it_message (const int counter, bool ks)
+{
+	if(ks){
+	        process_solution (counter);
+   	        std::cout << "iteration="<< counter+1 << "  J(u_h)= "<< phi_zero << ", H1 error: "
+        	<<  H1_error  <<", W0-1,infty error: "<< Linfty_error<< " J'(u_h)(w)= "<< phip
+       		<< ", |J'(u_h)|= "<< system_rhs.l2_norm()<<std::endl;
+			}
+	else {
+			std::cout << "iteration= " << counter+1 << " J(u_h)= " 
+			<< phi_alpha << " J'(u_h)= "<< phip<<std::endl; 
+		  } 
+}
+
+
+/*****************************************************************************************/
+//                            Convergence Tables
+
+
+/*************************************************************/
+// formating
+
+template <int dim>
+void ElastoplasticTorsion<dim>::format_convergence_tables()
+{
+			convergence_table.set_precision("L2", 3);
+			convergence_table.set_precision("H1", 3);
+			convergence_table.set_precision("Linfty", 3);
+			convergence_table.set_precision("function value", 3);
+			convergence_table.set_precision("derivative", 3);
+			dual_convergence_table.set_precision("dual_L1", 3);
+			dual_convergence_table.set_precision("dual_Linfty", 3);
+			dual_convergence_table.set_precision("L2", 3);
+			dual_convergence_table.set_precision("H1", 3);
+			dual_convergence_table.set_precision("Linfty", 3);
+			convergence_table.set_scientific("L2", true);
+			convergence_table.set_scientific("H1", true);
+			convergence_table.set_scientific("Linfty", true);
+			convergence_table.set_scientific("function value", true);
+			convergence_table.set_scientific("derivative", true);
+			dual_convergence_table.set_scientific("dual_L1", true);
+			dual_convergence_table.set_scientific("dual_Linfty", true);
+			dual_convergence_table.set_scientific("L2", true);
+			dual_convergence_table.set_scientific("H1", true);
+			dual_convergence_table.set_scientific("Linfty", true);
+
+}
+
+/****************************************/
+// fill-in entry for the solution
+template <int dim>
+void ElastoplasticTorsion<dim>::process_solution (const unsigned int it)
+{
+	Vector<float> difference_per_cell (triangulation.n_active_cells());
+	
+	// compute L2 error (save to difference_per_cell)
+	VectorTools::integrate_difference (dof_handler,present_solution,
+		Solution<dim>(),difference_per_cell,QGauss<dim>(3),VectorTools::L2_norm);
+	L2_error = difference_per_cell.l2_norm();
+
+	// compute H1 error (save to difference_per_cell)
+	VectorTools::integrate_difference (dof_handler,present_solution,Solution<dim>(),
+		difference_per_cell,QGauss<dim>(3),VectorTools::H1_seminorm);
+	H1_error = difference_per_cell.l2_norm();
+
+	// compute W1infty error (save to difference_per_cell)
+	const QTrapez<1> q_trapez;
+	const QIterated<dim> q_iterated (q_trapez, 5);
+	VectorTools::integrate_difference (dof_handler,present_solution,Solution<dim>(),
+	difference_per_cell,q_iterated,VectorTools::W1infty_seminorm);
+	Linfty_error = difference_per_cell.linfty_norm();
+
+
+	convergence_table.add_value("cycle", it);
+	convergence_table.add_value("p", p);
+	convergence_table.add_value("L2", L2_error);
+	convergence_table.add_value("H1", H1_error);
+	convergence_table.add_value("Linfty", Linfty_error);
+	convergence_table.add_value("function value", phi_alpha);
+	convergence_table.add_value("derivative", phip);
+}
+
+
+/***************************************/
+// fill-in entry  for the multiplier	
+template <int dim>
+void ElastoplasticTorsion<dim>::process_multiplier (const unsigned int cycle, const int iter,double time)
+{
+	const unsigned int n_active_cells=triangulation.n_active_cells();
+	const unsigned int n_dofs=dof_handler.n_dofs();
+	dual_L1_error=dual_error();
+	dual_L_infty_error=dual_infty_error();
+
+
+	dual_convergence_table.add_value("cycle", cycle);
+	dual_convergence_table.add_value("p", p);
+	dual_convergence_table.add_value("iteration_number", iter);
+	dual_convergence_table.add_value("cpu_time", time);
+	dual_convergence_table.add_value("cells", n_active_cells);
+	dual_convergence_table.add_value("dofs", n_dofs);
+	dual_convergence_table.add_value("L2", L2_error);
+	dual_convergence_table.add_value("H1", H1_error);
+	dual_convergence_table.add_value("Linfty", Linfty_error);
+	dual_convergence_table.add_value("dual_L1", dual_L1_error);
+	dual_convergence_table.add_value("dual_Linfty", dual_L_infty_error);
+
+}
+
+
+
+
+/****************************************************************************************/
+// ElastoplasticTorsion::setup_system
+// unchanged from step-15
+
+  template <int dim>
+  void ElastoplasticTorsion<dim>::setup_system (const bool initial_step)
+  {
+    if (initial_step)
+      {
+        dof_handler.distribute_dofs (fe);
+        present_solution.reinit (dof_handler.n_dofs());
+        grad_norm.reinit (dof_handler.n_dofs());
+        lambda.reinit (dof_handler.n_dofs());
+
+        hanging_node_constraints.clear ();
+        DoFTools::make_hanging_node_constraints (dof_handler,
+        hanging_node_constraints);
+        hanging_node_constraints.close ();
+      }
+
+
+    // The remaining parts of the function
+
+    newton_update.reinit (dof_handler.n_dofs());
+    system_rhs.reinit (dof_handler.n_dofs());
+    CompressedSparsityPattern c_sparsity(dof_handler.n_dofs());
+    DoFTools::make_sparsity_pattern (dof_handler, c_sparsity);
+    hanging_node_constraints.condense (c_sparsity);
+    sparsity_pattern.copy_from(c_sparsity);
+    system_matrix.reinit (sparsity_pattern);
+  }
+
+/***************************************************************************************/
+  /* the coeffcients W, W' and G defining the problem.
+   
+     Min_u \int W(|Du|^2) dx
+     
+  They must be consistent as G(s)=W'(s)+2s W''(s) for any s>0.
+  recall that they receive the SQUARED gradient. */
+
+  template <int dim>
+  double ElastoplasticTorsion<dim>::W (double Du2) const
+  {
+	return Du2;
+  }
+
+  template <int dim>
+  double ElastoplasticTorsion<dim>::Wp (double Du2) const
+  {
+	return 1.0;
+  }
+
+  template <int dim>
+  double ElastoplasticTorsion<dim>::G (double Du2) const
+  {
+	return 1.0;
+  }
+/***************************************************************************************/
+
+  template <int dim>
+  void ElastoplasticTorsion<dim>::assemble_system ()
+  {
+
+
+    const QGauss<dim>  quadrature_formula(3);
+	const RightHandSide<dim> right_hand_side;
+    system_matrix = 0;
+    system_rhs = 0;
+
+    FEValues<dim> fe_values (fe, quadrature_formula,
+                             update_gradients         |
+						      update_values           |
+                             update_quadrature_points |
+                             update_JxW_values);
+
+    const unsigned int           dofs_per_cell = fe.dofs_per_cell;
+    const unsigned int           n_q_points    = quadrature_formula.size();
+
+    FullMatrix<double>           cell_matrix (dofs_per_cell, dofs_per_cell);
+    Vector<double>               cell_rhs (dofs_per_cell);
+
+    std::vector<Tensor<1, dim> > old_solution_gradients(n_q_points);
+    std::vector<types::global_dof_index>    local_dof_indices (dofs_per_cell);
+
+
+    typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    endc = dof_handler.end();
+    for (; cell!=endc; ++cell)
+      {
+        cell_matrix = 0;
+        cell_rhs = 0;
+
+        fe_values.reinit (cell);
+        fe_values.get_function_gradients(present_solution,
+                                         old_solution_gradients);
+
+        for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
+          {
+             long double coeff=0.0;
+	     	 long double a=old_solution_gradients[q_point] * old_solution_gradients[q_point];	
+	     	 long double exponent=(p-2.0)/2*std::log(a);
+             coeff= std::exp( exponent);  
+             for (unsigned int i=0; i<dofs_per_cell; ++i)
+              {
+                for (unsigned int j=0; j<dofs_per_cell; ++j)
+                  { 
+                  	if (dir_id==1){
+                    	cell_matrix(i, j) +=  fe_values.shape_grad(i, q_point) *  fe_values.shape_grad(j, q_point)
+							* (G(a)+(p-1.0)*coeff)    * fe_values.JxW(q_point);
+					}
+					else {
+					    cell_matrix(i, j) +=  fe_values.shape_grad(i, q_point) *  fe_values.shape_grad(j, q_point)
+							* (Wp(a)+coeff)                                                  
+							   * fe_values.JxW(q_point);
+					}
+                  }
+
+                cell_rhs(i) -= (  fe_values.shape_grad(i, q_point)
+                                         * old_solution_gradients[q_point]
+											* (Wp(a)+coeff)
+                                 -right_hand_side.value(fe_values.quadrature_point(q_point))
+									*fe_values.shape_value(i, q_point)
+                                      ) 
+							* fe_values.JxW(q_point);
+              }
+           }
+
+		cell->get_dof_indices (local_dof_indices);
+        for (unsigned int i=0; i<dofs_per_cell; ++i)
+          {
+            for (unsigned int j=0; j<dofs_per_cell; ++j)
+              system_matrix.add (local_dof_indices[i],
+                                 local_dof_indices[j],
+                                 cell_matrix(i,j));
+
+            system_rhs(local_dof_indices[i]) += cell_rhs(i);
+          }
+    }
+
+    hanging_node_constraints.condense (system_matrix);
+    hanging_node_constraints.condense (system_rhs);
+
+    std::map<types::global_dof_index,double> boundary_values;
+    VectorTools::interpolate_boundary_values (dof_handler,
+                                              0,
+                                              ZeroFunction<dim>(),
+                                              boundary_values);
+    MatrixTools::apply_boundary_values (boundary_values,
+                                        system_matrix,
+                                        newton_update,
+                                        system_rhs);
+ }
+
+
+
+
+/**********************************      Refine Mesh      ****************************************/
+// unchanged from step-15
+
+  template <int dim>
+  void ElastoplasticTorsion<dim>::refine_mesh ()
+  {
+    Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
+    KellyErrorEstimator<dim>::estimate (dof_handler,
+                                        QGauss<dim-1>(3),
+                                        typename FunctionMap<dim>::type(),
+                                        present_solution,
+                                        estimated_error_per_cell);
+
+	prm.enter_subsection ("Mesh & Refinement Parameters");
+		const double top_fraction=prm.get_double("top_fraction_of_cells");
+		const double bottom_fraction=prm.get_double("bottom_fraction_of_cells");
+	prm.leave_subsection ();
+    GridRefinement::refine_and_coarsen_fixed_number (triangulation,
+                                                     estimated_error_per_cell,
+                                                     top_fraction, bottom_fraction);
+
+    triangulation.prepare_coarsening_and_refinement ();
+    SolutionTransfer<dim> solution_transfer(dof_handler);
+    solution_transfer.prepare_for_coarsening_and_refinement(present_solution);
+    triangulation.execute_coarsening_and_refinement();
+    dof_handler.distribute_dofs(fe);
+    Vector<double> tmp(dof_handler.n_dofs());
+    solution_transfer.interpolate(present_solution, tmp);
+    present_solution = tmp;
+    set_boundary_values ();
+    hanging_node_constraints.clear();
+
+    DoFTools::make_hanging_node_constraints(dof_handler,
+                                          hanging_node_constraints);
+    hanging_node_constraints.close();
+    hanging_node_constraints.distribute (present_solution);
+    setup_system (false);
+  }
+
+
+/*******************************************************************************************/
+// Dump the norm of the gradient and the lagrange multiplier in vtu format for visualization    
+  template <int dim>
+  void ElastoplasticTorsion<dim>::output_results (unsigned int counter) const
+  {    
+	// multiplier object contains both |Du| and lambda.
+	ComputeMultiplier<dim> multiplier(p);	
+	DataOut<dim> data_out;
+
+    data_out.attach_dof_handler (dof_handler);
+    data_out.add_data_vector (present_solution, "solution");
+    data_out.add_data_vector (present_solution, multiplier);
+    data_out.build_patches ();
+    std::ostringstream p_str;
+    p_str << p<<"-cycle-"<<counter;
+    std::string str = p_str.str();
+    const std::string filename = "solution-" + str+".vtu";
+    std::ofstream output (filename.c_str());
+    data_out.write_vtu (output);
+}
+
+/********************************************************************************************/
+// unchanged from step-15
+  template <int dim>
+  void ElastoplasticTorsion<dim>::set_boundary_values ()
+  {
+    std::map<types::global_dof_index, double> boundary_values;
+    VectorTools::interpolate_boundary_values (dof_handler,
+                                              0,
+                                              BoundaryValues<dim>(),
+                                              boundary_values);
+    for (std::map<types::global_dof_index, double>::const_iterator
+         bp = boundary_values.begin();
+         bp != boundary_values.end(); ++bp)
+      present_solution(bp->first) = bp->second;
+  }
+
+
+/****************************************************************************************/
+//  COMPUTE \phi(\alpha)=J_p(u_h+\alpha w) 
+  template <int dim>
+  double ElastoplasticTorsion<dim>::phi (const double alpha) const
+  {
+    double obj = 0.0;
+	const RightHandSide<dim> right_hand_side;
+    Vector<double> evaluation_point (dof_handler.n_dofs());
+    evaluation_point = present_solution;  // copy of u_h
+    evaluation_point.add (alpha, newton_update); // u_{n+1}=u_n+alpha w_n
+
+    const QGauss<dim>  quadrature_formula(3);
+    FEValues<dim> fe_values (fe, quadrature_formula,
+                             update_gradients         |
+		                     update_values    |
+                             update_quadrature_points |
+                             update_JxW_values);
+
+    const unsigned int           dofs_per_cell = fe.dofs_per_cell;
+    const unsigned int           n_q_points    = quadrature_formula.size();
+
+    Vector<double>               cell_residual (dofs_per_cell);
+    std::vector<Tensor<1, dim> > gradients(n_q_points);
+    std::vector<double> values(n_q_points);
+
+
+    std::vector<types::global_dof_index>    local_dof_indices (dofs_per_cell);
+
+    typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    endc = dof_handler.end();
+    for (; cell!=endc; ++cell)
+      {
+        cell_residual = 0;
+        fe_values.reinit (cell);
+        fe_values.get_function_gradients (evaluation_point, gradients);
+		fe_values.get_function_values (evaluation_point, values);
+
+
+        for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
+          {
+              double Du2=gradients[q_point] *  gradients[q_point]; // Du2=|Du|^2
+              double penalty;
+              if (Du2<1.0e-10)
+                penalty=0.0;
+              else
+                penalty=std::pow(Du2,p/2.0); // penalty=|Du|^p
+
+		      // obj+= 1/2 W(|Du|^2)+1/p |Du|^p -fu (see (1))
+              obj+=(    
+              	(0.5*W(Du2)+penalty/p)- right_hand_side.value(fe_values.quadrature_point(q_point))*values[q_point]
+                   ) * fe_values.JxW(q_point);
+            }
+
+        }
+
+    return obj;
+  }
+
+
+/***************************************************************************************************/
+// Compute L^1 error norm of Lagrange Multiplier
+// with respect to exact solution (cf. Alvarez & Flores, 2015)
+
+  template <int dim>
+  double ElastoplasticTorsion<dim>::dual_error () const
+  {
+    double obj = 0.0;
+
+    const QGauss<dim>  quadrature_formula(3);
+    FEValues<dim> fe_values (fe, quadrature_formula,
+                             update_gradients         |
+                             update_quadrature_points |
+                             update_JxW_values);
+
+    const unsigned int           dofs_per_cell = fe.dofs_per_cell;
+    const unsigned int           n_q_points    = quadrature_formula.size();
+
+    Vector<double>               cell_residual (dofs_per_cell);
+    std::vector<Tensor<1, dim> > gradients(n_q_points);
+  
+    std::vector<types::global_dof_index>    local_dof_indices (dofs_per_cell);
+
+    typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    endc = dof_handler.end();
+    for (; cell!=endc; ++cell)
+      {
+        cell_residual = 0;
+        fe_values.reinit (cell);
+        fe_values.get_function_gradients (present_solution, gradients);
+
+        for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
+          {
+              double coeff=gradients[q_point] *  gradients[q_point] ;
+              if (coeff<1.0e-15)
+                  coeff=0.0;
+              else
+                    coeff=std::pow(coeff,(p-2.0)/2.0); // |Du_p|^(p-2)
+
+              double r=std::sqrt(fe_values.quadrature_point(q_point).square());
+	      double exact=0;	      
+	      if (r>0.5)
+	      	exact= 2*r-1;
+		
+              obj+=(     std::abs(coeff-exact) ) * fe_values.JxW(q_point);
+            }
+
+        }
+
+    return obj;
+  }
+
+/*******************************************************************************************/
+// Compute L^infinity error norm of Lagrange Multiplier
+// with respect to exact solution (cf. Alvarez & Flores, 2015)
+
+  template <int dim>
+  double ElastoplasticTorsion<dim>::dual_infty_error () const
+  {
+    double obj = 0.0;
+   const QTrapez<1> q_trapez;
+   const QIterated<dim> quadrature_formula (q_trapez, 10);
+
+    FEValues<dim> fe_values (fe, quadrature_formula,
+                             update_gradients         |
+                             update_quadrature_points );
+
+    const unsigned int           dofs_per_cell = fe.dofs_per_cell;
+    const unsigned int           n_q_points    = quadrature_formula.size();
+
+    Vector<double>               cell_residual (dofs_per_cell);
+    std::vector<Tensor<1, dim> > gradients(n_q_points);
+  
+    std::vector<types::global_dof_index>    local_dof_indices (dofs_per_cell);
+
+    typename DoFHandler<dim>::active_cell_iterator
+    cell = dof_handler.begin_active(),
+    endc = dof_handler.end();
+    for (; cell!=endc; ++cell)
+      {
+        cell_residual = 0;
+        fe_values.reinit (cell);
+        fe_values.get_function_gradients (present_solution, gradients);
+
+        for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
+          {
+              long double sqdGrad=gradients[q_point] *  gradients[q_point] ;
+              double r=std::sqrt(fe_values.quadrature_point(q_point).square());
+	      double exact=0;	      
+	      if (r>0.5)
+	      	exact= 2*r-1.0;
+		// compute |Du|^(p-2) as  exp(p-2/2*log(Du^2)) 
+		long double exponent=(p-2.0)/2*std::log(sqdGrad);
+		long double coeff=std::exp(exponent);
+		
+             if(std::abs(coeff-exact)>obj )
+		obj=std::abs(coeff-exact);
+            }
+
+        }
+
+    return obj;
+  }
+
+/*****************************************************************************************/
+// check whether putative step-length satisfies sufficient decrease conditions
+  template <int dim>
+bool ElastoplasticTorsion<dim>::checkWolfe(double & alpha, double & phi_alpha) const
+{ 
+if (phi_alpha< phi_zero+line_search_tolerence*phip*alpha )
+	return true;
+   else
+	return false;		
+}
+
+
+/*****************************************************************************************/
+// Find a step-length satisfying sufficient decrease condition by line-search 
+// uses quadratic interpolation
+
+  template <int dim>
+bool ElastoplasticTorsion<dim>::determine_step_length(const int inner_it) 
+{
+    unsigned int it=0;
+	bool done;
+	double alpha,nalpha;
+	prm.enter_subsection ("Algorithm Parameters");
+		const unsigned int max_LS_it=prm.get_integer("max_LS_it");
+		double init_SL=prm.get_double("init_step_length");
+	prm.leave_subsection ();
+	if (inner_it==0)
+		alpha=init_SL;
+	else
+	{
+		alpha=std::min(1.45*old_step*old_phip/phip,1.0);
+	}
+	phi_alpha=phi(alpha);
+	std::cerr << "Step length=" << alpha << ", Value= " << phi_alpha;
+	// check if step-size satisfies sufficient decrease condition
+    done=checkWolfe(alpha,phi_alpha);
+    if (done)
+    		std::cerr << " accepted" << std::endl;
+	else 
+		std::cerr << " rejected" ;
+		
+	while ((!done) & (it<max_LS_it)) {
+		// new try obtained by quadratic interpolation
+		nalpha=-(phip*alpha*alpha)/(2*(phi_alpha-phi_zero-phip*alpha));
+
+        if (nalpha<1e-3*alpha ||  std::abs(nalpha-alpha)/alpha<1e-8)  
+          nalpha=alpha/2;
+		else if( phi_alpha-phi_zero>1e3*std::abs(phi_zero) )
+			nalpha=alpha/10;
+		alpha=nalpha;
+		phi_alpha=phi(alpha);
+		done=checkWolfe(alpha,phi_alpha);
+		if (done)
+			std::cerr << ", finished with steplength= "<< alpha<< ", fcn value= "<< phi_alpha<<std::endl;
+		it=it+1;
+	}
+	if (!done){
+		std::cerr << ", max. no. of iterations reached wiht steplength= "<< alpha
+		<< ", fcn value= "<< phi_alpha<<std::endl;
+		return false;
+	}
+	else{
+		  step_length=alpha;		
+		  return true;
+	}
+
+}
+
+/**************************************************************************************************/
+  // ElastoplasticTorsion::init_mesh()
+
+  template <int dim>
+  void ElastoplasticTorsion<dim>::init_mesh ()
+  {
+	// get parameters
+	prm.enter_subsection ("Mesh & Refinement Parameters");
+		const int domain_id=prm.get_integer("Code for the domain");
+		const int init_ref=prm.get_integer("No of initial refinements");
+	prm.leave_subsection ();
+	
+
+	if (domain_id==0){
+    // For the unit disk around the origin
+    GridGenerator::hyper_ball (triangulation);
+    static const HyperBallBoundary<dim> boundary;
+    triangulation.set_boundary (0, boundary);
+					 }
+	else if (domain_id==1){
+	// For the unit square
+    GridGenerator::hyper_cube (triangulation, 0, 1);}
+	else if (domain_id==2){
+	/* For Glowinski's domain
+      ___    ___   __ 1
+     |   |__|   |  __ .8
+	 |	        |
+	 |          |
+	 |__________|  __ 0 
+
+     |   |  |   |
+     0  .4 .6   1 
+	    
+	*/
+	Triangulation<dim> tria1;
+	Triangulation<dim> tria2;
+	Triangulation<dim> tria3;
+	Triangulation<dim> tria4;
+	Triangulation<dim> tria5;
+	Triangulation<dim> tria6;
+	GridGenerator::hyper_rectangle(tria1, Point<2>(0.0,0.0), Point<2>(0.4,0.8));
+	GridGenerator::hyper_rectangle(tria2, Point<2>(0.0,0.8), Point<2>(0.4,1.0));
+	GridGenerator::hyper_rectangle(tria3, Point<2>(0.4,0.0), Point<2>(0.6,0.8));
+	GridGenerator::hyper_rectangle(tria4, Point<2>(0.6,0.0), Point<2>(1.0,0.8));
+	GridGenerator::hyper_rectangle(tria5, Point<2>(0.6,0.8), Point<2>(1.0,1.0));
+	GridGenerator::merge_triangulations (tria1, tria2, tria6);
+	GridGenerator::merge_triangulations (tria6, tria3, tria6);
+	GridGenerator::merge_triangulations (tria6, tria4, tria6);
+	GridGenerator::merge_triangulations (tria6, tria5, triangulation);
+							}
+	// perform initial refinements
+    triangulation.refine_global(init_ref);
+}
+
+/**************************************************************************************************/
+  // ElastoplasticTorsion::solve(inner_it)
+  // Performs one inner iteration
+
+  template <int dim>
+  bool ElastoplasticTorsion<dim>::solve (const int inner_it)
+  {
+	prm.enter_subsection ("Algorithm Parameters");
+		const unsigned int max_CG_it=prm.get_integer("Max_CG_it");
+		const double CG_tol=prm.get_double("CG_tol");
+	prm.leave_subsection ();
+
+    SolverControl solver_control (max_CG_it,CG_tol);
+    SolverCG<>    solver (solver_control);
+
+    PreconditionSSOR<> preconditioner;
+    preconditioner.initialize(system_matrix,0.25);
+
+    solver.solve (system_matrix, newton_update, system_rhs,
+                  preconditioner);
+    hanging_node_constraints.distribute (newton_update);
+   /******  save current quantities for line-search  **** */
+   // Recall that phi(alpha)=J(u+alpha w)
+    old_step=step_length;	
+    old_phi_zero=phi_zero;
+    phi_zero=phi(0); // phi(0)=J(u)
+    old_phip=phip;
+    phip=-1.0*(newton_update*system_rhs); //phi'(0)=J'(u) *w, rhs=-J'(u).
+	if (inner_it==0)
+		phip_zero=phip;
+
+    if (phip>0){	// this should not happen, step back
+		 std::cout << "Not a descent direction!" <<std::endl;
+    	present_solution.add (-1.0*step_length, newton_update);
+		step_length=step_length/2;
+		phip=old_phip;
+		return false;
+					}
+	else{
+		if(determine_step_length(inner_it)){
+			// update u_{n+1}=u_n+alpha w_n
+		 	present_solution.add (step_length, newton_update);
+		 	return true;}
+		 	else return false;
+		}
+}
+
+
+
+/*************************************************************************************************************/
+// ElastoplasticTorsion::run
+template <int dim>
+void ElastoplasticTorsion<dim>::run ()
+{
+
+	// get parameters
+	prm.enter_subsection ("Mesh & Refinement Parameters");
+		const int adapt_ref=prm.get_integer("No of adaptive refinements");
+	prm.leave_subsection ();
+	prm.enter_subsection ("Algorithm Parameters");
+		const int max_inner=prm.get_integer("Max_inner");
+		const double eps=prm.get_double("eps");
+		const double hi_eps=prm.get_double("hi_eps");
+		const int hi_th=prm.get_integer("hi_th");
+		const double init_p=prm.get_double("init_p");
+		const double delta_p=prm.get_double("delta_p");
+	prm.leave_subsection ();
+	prm.enter_subsection ("Global Parameters");
+		bool known_solution=prm.get_bool("known_solution");
+		double actual_p=prm.get_double("p");
+	prm.leave_subsection ();
+	/************************/
+
+	// init Timer
+    Timer timer;
+    double ptime=0.0;
+    timer.start ();
+    	
+	// initalize mesh for the selected domain
+	init_mesh();
+
+	// setup FE space
+    setup_system (true);
+    set_boundary_values ();
+
+	// init counters
+	int global_it=0;		// Total inner iterations (counting both loops)
+	int cycle=0; 	        // Total outer iterations (counting both loops)
+	int refinement = 0;    //  Refinements  performed (adaptive) = outer iterations 2nd loop
+
+  
+    // prepare to start first loop  
+	p=init_p;	
+	bool well_solved=true;
+
+	/*****************************          First loop      ***********************************/
+	/****************** Prepare initial condition using increasing p  *************************/
+	while(p<actual_p) // outer iteration, increasing p.
+	{
+    	std::cout <<"--Preparing initial condition with p="<<p<<" iter.= " << global_it<<  "  .-- "<< std::endl;
+	    timer.restart(); 
+        for (int inner_iteration=0; inner_iteration<max_inner; ++inner_iteration,++global_it)
+		{
+			assemble_system ();
+	        well_solved=solve (inner_iteration);
+	        print_it_message (global_it, known_solution);
+            if(
+	            ((system_rhs.l2_norm()/std::sqrt(system_rhs.size()) <1e-4) & (cycle<1)) |
+    	        ((system_rhs.l2_norm()/std::sqrt(system_rhs.size()) <1e-5) & (cycle>=1)) |
+    	        !well_solved
+               )
+               break;			
+		}
+		ptime=timer();
+		if (well_solved)
+			output_results (cycle);
+		    		  
+        if(known_solution){
+     		process_multiplier(cycle,global_it,ptime);
+			//dual_convergence_table.write_tex(dual_error_table_file);
+		}
+        refine_mesh();
+        cycle++;  
+        p+=delta_p;
+    }
+    /***************************    first loop finished        ********************/
+    
+    
+    // prepare for second loop      
+	p=actual_p;
+	well_solved=true;
+	
+	
+	/*****************************        Second loop         *********************************/
+	/**************************** Solve problem for target p  *********************************/
+	
+    std::cout << "============ Solving  problem with p="   <<p << " =================="  << std::endl;
+    /*****    Outer iteration - refining mesh  ******************/
+    while ((cycle<adapt_ref) & well_solved) 
+      {
+		timer.restart();
+		// inner iteration
+        for (int inner_iteration=0; inner_iteration<max_inner; ++inner_iteration,++global_it)
+          {	   
+            assemble_system ();
+            well_solved=solve (inner_iteration);
+            print_it_message (global_it, known_solution);
+            
+            if( 
+               ((system_rhs.l2_norm()/std::sqrt(system_rhs.size()) < eps) & (refinement<hi_th)) | 
+               (( system_rhs.l2_norm()/	std::sqrt	(system_rhs.size()) <hi_eps)  | (!well_solved))
+               )
+               break;			
+          }
+          //inner iterations finished
+        ptime=timer();
+   		if (well_solved)
+			output_results (cycle);
+	    
+		// compute and display error, if the explicit solution is known
+		if(known_solution){
+			process_multiplier(cycle,global_it,ptime);
+			std::cout << "finished with H1 error: " <<  H1_error << ", dual error (L1): "
+			<< dual_L1_error << "dual error (L infty): "<<dual_L_infty_error <<std::endl;
+		} 
+
+		// update counters						
+		++refinement;
+		++cycle;
+		// refine mesh	
+        std::cout << "******** Refined mesh " << cycle    << " ********"  << std::endl;
+        refine_mesh();
+	}// second loop
+
+	// write convergence tables to file
+	if(known_solution){
+		format_convergence_tables();	
+		std::string error_filename = "error"+Method+elements+".tex";
+		std::ofstream error_table_file(error_filename.c_str());
+		std::string dual_error_filename = "dual_error"+Method+elements+".tex";
+		std::ofstream dual_error_table_file(dual_error_filename.c_str());
+		convergence_table.write_tex(error_table_file);
+		dual_convergence_table.write_tex(dual_error_table_file);
+	}
+  }//run()
+
+}//namespace
+
+/**********************************************************************************************/
+// The main function
+int main ()
+{
+	try
+    {
+		using namespace dealii;
+		using namespace nsp;
+		deallog.depth_console (0);
+
+		ParameterHandler prm;
+		ParameterReader param(prm);
+		param.read_parameters("EPT.prm");
+		ElastoplasticTorsion<2> ElastoplasticTorsionProblem(prm);
+		ElastoplasticTorsionProblem	.run ();
+    }
+  catch (std::exception &exc)
+    {
+      std::cerr << std::endl << std::endl
+                << "----------------------------------------------------"    << std::endl;
+      std::cerr << "Exception on processing: " << std::endl
+                << exc.what() << std::endl
+                << "Aborting!" << std::endl
+                << "----------------------------------------------------"
+                << std::endl;
+
+      return 1;
+    }
+  catch (...)
+    {
+      std::cerr << std::endl << std::endl
+                << "----------------------------------------------------"
+                << std::endl;
+      std::cerr << "Unknown exception!" << std::endl
+                << "Aborting!" << std::endl
+                << "----------------------------------------------------"
+                << std::endl;
+      return 1;
+    }
+  return 0;
+}
diff --git a/Elastoplastic Torsion/doc/author b/Elastoplastic Torsion/doc/author
new file mode 100644
index 0000000..8ff862d
--- /dev/null
+++ b/Elastoplastic Torsion/doc/author	
@@ -0,0 +1 @@
+Salvador Flores <sflores@dim.uchile.cl>
diff --git a/Elastoplastic Torsion/doc/builds-on b/Elastoplastic Torsion/doc/builds-on
new file mode 100644
index 0000000..f54144f
--- /dev/null
+++ b/Elastoplastic Torsion/doc/builds-on	
@@ -0,0 +1 @@
+step-15, step-29.